改进的稀疏近似逆预条件算法求解电磁场边值问题

提出了一种MAINV稀疏近似逆预条件算法,用于改善电磁场边值问题的有限元分析所产生的的线性系统的迭代求解。该预条件子是在基本AINV算法基础上,在分解过程中对可能导致算法崩溃的极小主元进行实时补偿,从而获得高质量的预条件子。数值结果表明,MAINV预条件子对SQMR以及若干经典迭代法的加速效果十分明显;此外,与其他常规预条件子相比较,MAINV具有更好的求解性能。     

基于COCG的组合迭代法求解电磁场FEM线性方程组

针对COCG算法在求解有限元法分析电磁场边值问题所得到的线性系统时存在的不稳定性问题,提出了一种基于COCG的组合迭代方法。该方法通过在低精度迭代阶段采用COCG,而在其易出现迭代崩溃问题的高精度迭代阶段采用其他相对更稳定的迭代法来进行联合求解。该方法不仅能有效避免COCG算法的不稳定性问题,同时也让其高效率的优点得到充分保留因而有利于整个迭代收敛过程。数值结果表明,提出的组合迭代法能将COCG的高效率和其他迭代法的稳定性进行优势互补,从而能获得比常规的单独迭代法更高的求解性能。     

求解Helmholtz方程的新型稀疏近似逆预条件算法

提出了一种新型预条件算法,用于对有限元法离散Helmholtz方程所产生的大型稀疏复对称且高度不定的线性系统进行高效迭代求解。该新型预条件子是在复拉普拉斯偏移算子的基础上结合改进的稀疏近似逆算法来得到。通过改善矢量有限元线性系统自身的谱特性,该预条件算法既可避免迭代中的不稳定情况,同时也能较大提高迭代求解效率。数值结果表明,与若干常用预条件算法相比,所提出的预条件算法更加有效。     

光电器件仿真分析中的COCG高效迭代算法

对光电器件采用FEM/EFIE仿真分析所产生的线性系统的迭代求解算法进行了研究。与目前普遍使用的迭代法不同,针对FEM/EFIE系数矩阵的特点,提出了采用求解复对称且非正定的线性方程组的共轭正交共轭梯度（COCG）算法来进行高效迭代求解。数值实验基于对波导元件分别采用矢量有限元法（FEM）和电场积分方程法（EFIE）得到的两类典型线性系统进行迭代求解。结果表明：与常规迭代法相比,COCG在求解速度和内存使用上的性能优势非常明显,从而能较大地提高仿真效率。     

求解复杂载体天线辐射问题的近场预条件技术

提出了一种近场预条件技术与LDU分解法相结合的新技术,用于加速矩量法(MoM)分析复杂载体上线天线辐射问题中线性方程组的迭代求解.通过LDU分解可将系数矩阵中表示载体上单元相互作用的具有对角占优特性的子阵分离出来,构造一个矩阵分解形式的预条件阵.结合广义最小留数(GMRES)法,分别对装载在两个简单形体和一架大型飞机模型上的线天线的辐射问题进行了求解.数值结果表明,该方法可大大加快线性方程组迭代求解的收敛速度,提高分析计算效率.     

正则化预条件方法在矩量法中的应用

计算电磁学中矩量法产生的系统矩阵是病态矩阵,使用迭代方法求解时很难收敛,即使采用现有的预条件技术也经常不收敛.本文借用不适定问题求解中的正则化方法的概念,提出采用正则化矩阵作为矩量法中矩阵方程的一个预条件矩阵.这种预条件方法可以直接改善原矩阵的特征值分布,而且不需要额外的空间来存储预条件矩阵.此外,本文提出通过正则化矩阵方程的L曲线的二阶导数的最大值点来确定正则化参数,使得预条件矩阵方程求解的效率最高.数值实验表明,对于高阶矩量法求解电场积分方程或者磁场积分方程时分别产生的矩阵方程,采用常见的预条件迭代方法求解时收敛很慢,但是采用本文的预条件迭代方法却可以较快地收敛.     

多层低频快速多极子算法分析电磁散射问题

主要研究了低频条件下目标的散射问题,详细给出了基于Loop-Tree矩量法的多层低频快速多极子方法基本原理。通过对自由空间格林函数进行多极子展开,避免了传统快速多极子方法通过平面波展开格林函数遇到的低频崩溃问题。改进的对角预条件技术显著地减少了预条件矩阵的构造时间和矩阵的迭代求解时间。数值算例证明了算法的有效性。     

基于MT-SIE法求解电大介质目标电磁散射

介绍了一种用于均匀介质目标电磁散射求解的新型多区域表面积分方程(MT-SIE)方法。不同于传统的用于介质目标散射求解的积分方法,该方法将均匀介质目标分解为内、外2个独立的子区,通过在介质表面强加Robin传输条件来保证电流和磁流的连续性。由于介质目标被分解为内外2个独立的子区,不同的子区允许非共形剖分。相较于传统方法,该方法可以更高效地与多层快速多级子(MLFMA)相结合求解电大尺寸目标。为进一步加速矩阵的迭代求解,提出了一种高斯-赛德尔型预条件技术,可以有效改善矩阵的收敛,加快迭代求解速度。     

邻居预条件加速的多层快速非均匀平面波算法

采用邻居预条件加速的多层快速非均匀平面波算法求解三维导电目标的电磁散射.通过分组,将耦合划分为附近和非附近区,对于非附近区采用索末菲恒等式对格林函数展开,用修正最陡下降路径代替索末菲积分路径进行数值积分.采用内插与外推技术将复角谱序列转换成均匀实角谱序列,以便于算法的高效实施.该算法的计算复杂度与多层快速多极子相当,且更具潜在优势.为改善迭代特性,本文研究了一种邻居预条件方法,加速迭代收敛,数值结果验证了算法的准确和高效.     

并行多层快速多极子的高效预条件技术

多层快速多极子法是基于矩量法的快速算法,具有较低的计算复杂度和存储复杂度,被广泛应用于目标电磁散射特性分析。对于复杂结构电磁目标,由于矩阵条件数较差,往往存在迭代收敛慢甚至不收敛的问题。针对这一情况,文中利用快速多极子的近区矩阵,结合稀疏矩阵方程求解构造了一种高效预条件。数值实例表明该方法相比于块对角预条件效果更好,能有效加速多层快速多极子迭代过程。     

An Efficient MAINV Preconditioned COCG Method for FEM Analysis of Millimeter Wave Filters

The modified AINV (MAINV) sparse approximate inverse preconditioner is applied to the conjugate orthogonal conjugate gradient (COCG) iterative method for solving a large systems of linear equations resulting from the use of edge finite element method (FEM). The proposed preconditioner is derived from basic AINV process by adding pivots compensation strategy to avoid the potential breakdowns. Numerical experiments on several typical millimeter wave structrues demonstrate the effectiveness of the MAINV-COCG method, in comparison with other conventional methods.     

一种堆叠式3D IC的最小边界热分析方法

目前的热分析工具仅仅支持单芯片的热分析,而堆叠式的三维芯片(3D IC)在同一封装中包含多个堆叠的芯片,对芯片的散热和温度管理提出了更高的要求,并且在热分析过程中需要处理复杂的边界条件.本文提出的最小边界法可以准确且有效地处理堆叠式3D IC的边界条件,简化了三维芯片封装的热模型;同时,本文提出在堆叠式3D IC的稳态热量分析中通过将连接点分类、采用预处理矩阵的方法加速整个全局热传导矩阵的求解过程,从而简化热分析流程.实验结果表明:将有限元方法作为基本的热分析方法,用最小边界法处理堆叠式3D IC,可以准确分析芯片的热分布;同时通过高效的预处理矩阵可以减少共轭梯度法求解中90%的迭代次数.     

Spectral Two-Step Preconditioning of Multilevel Fast Multipole Algorithm for the Fast Monostatic RCS Calculation

A new spectral two-step preconditioning of multilevel fast multipole algorithm (MLFMA) is proposed to solve large dense linear systems with multiple right-hand sides arising in monostatic radar cross section (RCS) calculations. The first system is solved with a deflated generalized minimal residual (GMRES) method and the eigenvector information is generated at the same time. Based on this eigenvector information, a spectral preconditioner is defined and combined with a previously constructed sparse approximate inverse (SAI) preconditioner in a two-step manner, resulting in the proposed spectral two-step preconditioner. Restarted GMRES with the newly constructed spectral two-step preconditioner is considered as the iterative method for solving subsequent systems and the MLFMA is used to speed up the matrix-vector product operations. Numerical experiments indicate that the new preconditioner is very effective with the MLFMA and can reduce both the iteration number and the computational time significantly.     

Reduced-order preconditioning for bidomain simulations

Simulations of the bidomain equations involve solving large, sparse, linear systems of the form Ax = b. Being an initial value problems, it is solved at every time step. Therefore, efficient solvers are essential to keep simulations tractable. Iterative solvers, especially the preconditioned conjugate gradient (PCG) method, are attractive since memory demands are minimized compared to direct methods, albeit at the cost of solution speed. However, a proper preconditioner can drastically speed up the solution process by reducing the number of iterations. In this paper, a novel preconditioner for the PCG method based on system order reduction using the Arnoldi method (A-PCG) is proposed. Large order systems, generated during cardiac bidomain simulations employing a finite element method formulation, are solved with the A-PCG method. Its performance is compared with incomplete LU (ILU) preconditioning. Results indicate that the A-PCG estimates an approximate solution considerably faster than the ILU, often within a single iteration. To reduce the computational demands in terms of memory and run time, the use of a cascaded preconditioner was suggested. The A-PCG was applied to quickly obtain an approximate solution, and subsequently a cheap iterative method such as successive overrelaxation (SOR) is applied to further refine the solution to arrive at a desired accuracy. The memory requirements are less than those of direct LU but more than ILU method. The proposed scheme is shown to yield significant speedups when solving time evolving systems.     

A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering

A highly effective preconditioner is presented for solving the system of equations obtained from the application of the hybrid finite element-boundary integral (FE-BI) method to three-dimensional (3-D) electromagnetic scattering problems. Different from widely used algebraic preconditioners, the proposed one is based on a physical approximation and is constructed from the finite element method (FEM) using an absorbing boundary condition (ABC) on the truncation boundary. It is shown that the large eigenvalues of the finite element (FE)-ABC system are similar to those of the FE-BI system. Hence, the preconditioned system has a spectrum distribution clustered around 1 in the complex plane. Consequently, when a Krylov subspace based method is employed to solve the preconditioned system, the convergence can be greatly accelerated. Numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems.     

Parallel multigrid preconditioner for the cardiac bidomain model

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.